Continuum analyses of structures containing cracks
The previous chapters of this book have demonstrated how multi-scale models and experiments at very small scales have improved and offer promise for furthering our understanding of the mechanical behavior and physical properties of biological materials an
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Introduction
The previous chapters of this book have demonstrated how multi-scale models and experiments at very small scales have improved and offer promise for furthering our understanding of the mechanical behavior and physical properties of biological materials and structures. This chapter illustrates how valuable insights can be gained using simplified continuum mechanics models that gloss over the complexities that can be handled by the multiscale models, such as numerous distinct nano and micro scale features that comprise the structures, material inhomogeneity, biological and chemical processes, the precise geometrical description, and the presence of flaws. This as long as the simplified treatments capture the essential features controlling the mechanical response. The reward for sacrificing the details is that the simplified models are amenable to analytical treatment. The discussion is limited to linear elastic structures containing cracks. The fracture mechanics analyses rely on the concepts associated with the energy release rate (ERR), the universal form of the stress and displacement components in the vicinity of the crack front, the generalized Griffith crack growth criterion for small-scale-yielding (SSY) conditions, and stability of crack propagation under displacement and load controls. Using numerous examples the point will be driven home that if one can accurately estimate the changes in stored elastic energy produced by variations in the crack length, then conditions for crack extension and the stability of crack propagation can be established. The ERR represents the variation in strain energy, or alternatively in potential energy resulting from crack extension. It is often referred to as the crack driving force because as discussed subsequently a crack will extend only if the ERR is sufficiently large to overcome the material’s resistance to the creation of the additional crack surfaces. In addition, for the perfectly brittle materials considered here the stability of crack propagation is determined by the change in the ERR with respect to a unit increase in crack M. Buehler, R. Ballarini (Eds.), Materiomics: Multiscale Mechanics of Biological Materials and Structures, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1574-9_7, © CISM, Udine 2013
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surface area. All derivations and examples are for planar and traction-free cracks that grow along their defining plane. Two definitions of ERR are provided first, both motivated from the load-displacement curves of a structure containing cracks having slightly different lengths. Selected examples are then presented and solved that elucidate how accurate approximations of the ERR can be achieved for certain types of crack configurations using simplified stress analysis procedures. This is followed by relating the ERR to the singular stress field in the immediate vicinity of the crack front (in terms of the to-be defined stress intensity factors) and to the global stiffness of the cracked component.
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Energy approac
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